Advances in Evolutionary Computing pp 3-44 | Cite as
Smoothness, Ruggedness and Neutrality of Fitness Landscapes: from Theory to Application
Abstract
The theory of fitness landscapes has been developed to provide a suitable mathematical framework for studying the evolvability of a variety of complex systems. In evolutionary computation the notion of evolvability refers to the efficiency of evolutionary search. It has been shown that the structure of a fitness landscape affects the ability of evolutionary algorithms to search. Three characteristics specify the structure of landscapes. These are the landscape smoothness, ruggedness and neutrality. The interplay of these characteristics plays a vital role in evolutionary search. This has motivated the appearance of a variety of techniques for studying the structure of fitness landscapes. An important feature of these techniques is that they characterize the landscapes by their smoothness and ruggedness, ignoring the existence of neutrality. Perhaps, the reason for this is that the role of neutrality in evolutionary search is still poorly understood.
Keywords
Autocorrelation Function Amplitude Spectrum Circuit Evolution Fitness Landscape Digital CircuitPreview
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References
- 1.Wright, S. (1932) The roles of mutation, inbreeding, crossbreeding and selection in evolution. In Jones, D.F., ed.: Proceedings of the 6th International Conference on Genetics. 1, 356–366Google Scholar
- 2.Davidor, Y. (1990) Epistasis variance: Suitability of a representation to genetic algorithms. Complex Systems. 4, 369–383Google Scholar
- 3.Davidor, Y. (1991) Epistasis variance: A viewpoint on ga-hardness. In Rawlins, G.J.E., ed.: Foundations of Genetic Algorithms. Morgan Kaufmann, San Mateo, CA, 23–35Google Scholar
- 4.Goldberg, D. (1987) Simple genetic algorithms and the minimal deceptive problem. In Davis, L., ed.: Genetic Algorithms and Simulated Annealing. Pitman, London, 74–88Google Scholar
- 5.Goldberg, D. (1989) Genetic algorithms and Walsh functions: Part I, a gentle introduction. Complex Systems. 3, 129–152MathSciNetzbMATHGoogle Scholar
- 6.Goldberg, D. (1989) Genetic algorithms and Walsh functions: Part II, deception and its analysis. Complex Systems. 3, 153–171MathSciNetzbMATHGoogle Scholar
- 7.Whitley, D.L. (1991) Fundamental principles of deception in genetic search. In Rawlins, G.J.E., ed.: Foundations of Genetic Algorithms. Morgan Kaufmann, San Mateo, CA, 221–241Google Scholar
- 8.Whitley, D.L. (1992) Deception, dominance and implicit parallelism in genetic search. Annals of Mathematics and Artificial Intelligence. 5, 49–78MathSciNetzbMATHCrossRefGoogle Scholar
- 9.Altenberg, L. (1995) The schema theorem and price’s theorem. In Whitley, L.D., Vose, M.D., eds.: Foundations of Genetic Algorithms, Volume 3. Morgan Kaufmann, San Francisco, CA, 23–49Google Scholar
- 10.Horn, J., Goldberg, D. (1995) Genetic algorithm difficulty and the modality of fitness landscapes. In Whitley, L.D., Vose, M.D., eds.: Foundations of Genetic Algorithms, volume 3. Morgan Kaufmann, San Francisco, CA, 243–269Google Scholar
- 11.Palmer, R. (1991) Optimization on rugged landscapes. In Perelson, A., Kauffman, S., eds.: Molecular Evolution on Rugged Landscapes. Volume IX of SFI Studies in the Sciences of Complexity. Addison-Wesley, Reading, MA, 3–25Google Scholar
- 12.Kauffman, S. (1989) Adaptation on rugged fitness landscapes. In Stein, D., ed.: Lectures in the Sciences of Complexity. SFI Studies in the Sciences of Complexity. Addison-Wesley, Reading, MA, 527–618Google Scholar
- 13.Weinberger, E.D. (1990) Correlated and uncorrelated fitness landscapes and how to tell the difference. Biological Cybernetics. 63, 325–336zbMATHCrossRefGoogle Scholar
- 14.Stadler, P.F., Happel, R. (1992) Correlation structure of the landscape of the graph-bipartitioning problem. J. Phys. A: Math. Gen. 25, 3103–3110MathSciNetCrossRefGoogle Scholar
- 15.Stadler, P.F., Schnabl, W. (1992) The landscape of the traveling salesman problem. Physical Letters A. 161, 337–344MathSciNetzbMATHCrossRefGoogle Scholar
- 16.Manderick, B., de Weger, M., Spiessens, P. (1991) The genetic algorithm and the structure of the fitness landscape. In Belew, R.K., Booker, L.B., eds.: Proceedings of the 4th International Conference on Genetic Algorithms, San Mateo, CA, Morgan Kaufmann, 143–150Google Scholar
- 17.Mitchell, M., Forrest, S., Holland, J. (1991) The royal road for genetic algorithms: Fitness landscapes and ga performance. In Varela, J., Bourgine, P., eds.: Proceedings of the 1st European Conference on Artificial Life, Cambridge, MA, MIT Press, 245–254Google Scholar
- 18.Mathias, K., Whitley, D. (1992) Genetic operators, the fitness landscape and the traveling salesman problem. In Männer, R., Manderick, B., eds.: Parallel Problem Solving from Nature II, North-Holland, Elsevier Science Publishers, 219–228Google Scholar
- 19.Jones, T. (1995) Evolutionary Algorithms, Fitness Landscapes and Search. PhD thesis, University of New Mexico, Albuquergue, NMGoogle Scholar
- 20.Gitchoff, P., Wagner, G.P. (1996) Recombination induced hypergraphs: A new approach to mutation-recombination isomorphism. Complexity. 2, 37–43MathSciNetCrossRefGoogle Scholar
- 21.Culberson, J.C. (1995) Mutation-crossover isomorphism and the construction of discriminating functions. Evolutionary Computation. 2, 279–311CrossRefGoogle Scholar
- 22.Stadler, P.F. (1995) Towards theory of landscapes. In Lop éz-Pena, R., Capovilla, R., Garcia-Pelayo, R., Waelbroeck, H., Zertuche, F., eds.: Complex Systems and Binary Networks. Springer-Verlag, Berlin, 77–163Google Scholar
- 23.Stadler, P.F., Wagner, G.P. (1997) Algebraic theory of recombination spaces. Evolutionary Computation. 5, 241–275CrossRefGoogle Scholar
- 24.Wagner, G.P., Stadler, P.F. (1998) Complex adaptations and the structure of recombination spaces. In Nehaniv, C., Ito, M., eds.: Algebraic Engineering. World Scientific, Singapore, 96–115Google Scholar
- 25.Stadler, P.F. (1996) Landscapes and their correlation functions. J. Math. Chem. 20, 1–45MathSciNetzbMATHCrossRefGoogle Scholar
- 26.Hordijk, W., Stadler, P.F. (1998) Amplitude spectra of fitness landscapes. Adv Complex Systems. 1, 39–66zbMATHCrossRefGoogle Scholar
- 27.Vassilev, V.K. (1997) An information measure of landscapes. In Bäck, T., ed.: Proceedings of the 7th International Conference on Genetic Algorithms, San Francisco, CA, Morgan Kaufmann, 49–56Google Scholar
- 28.Vassilev, V.K., Fogarty, T.C., Miller, J.F. (2000) Information characteristics and the structure of landscapes. Evolutionary Computation 8 In press.Google Scholar
- 29.Higuchi, T., Niwa, T., Tanaka, T., Iba, H., de Garis, H., Furuya, T. (1992) Evolving hardware with genetic learning. In: Proceedings of Simulation of Adaptive Behaviour, Cambridge, MA, MIT Press, 417–424Google Scholar
- 30.Higuchi, T., Iwata, M., eds. (1996) Proceedings of the 1st International Conference on Evolvable Systems: From Biology to Hardware. Volume 1259 of Lecture Notes in Computer Science., Berlin, Springer-VerlagGoogle Scholar
- 31.Tomassini, M., Sanchez, E., eds. (1996) Towards Evolvable Hardware: The Evolutionary Engineering Approach. Volume 1062 of Lecture Notes in Computer Science, Springer-Verlag, BerlinGoogle Scholar
- 32.Mange, D., Tomassini, M., eds. (1998) Bio-Inspired Computing Machines: Towards Novel Computational Architectures Presses Polytechniques et Universitaires RomandesGoogle Scholar
- 33.Thompson, A. (1998) Hardware Evolution: Automatic Design of Electronic Circuits in Reconfigurable Hardware by Artificial Evolution. Springer-Verlag, LondonGoogle Scholar
- 34.Miller, J.F., Job, D., Vassilev, V.K. (2000) Principles in the evolutionary design of digital circuits — part i. Journal of Genetic Programming and Evolvable Machines 1 In press.Google Scholar
- 35.Miller, J.F., Thomson, P. (1998) Aspects of digital evolution: Geometry and learning. In Sipper, M., Mange, D., P érez-Uribe, A., eds.: Proceedings of the 2nd International Conference on Evolvable Systems: From Biology to Hardware. Volume 1478 of Lecture Notes in Computer Science, Springer-Verlag, Heidelberg, 25–35CrossRefGoogle Scholar
- 36.Miller, J.F., Thomson, P. (1998) Aspects of digital evolution: Evolvability and architecture. In Eiben, A.E., Bäck, T., Schoenauer, M., Schwefel, H.P., eds.: Parallel Problem Solving from Nature V. Volume 1498 of Lecture Notes in Computer Science. Springer, Berlin, 927–936CrossRefGoogle Scholar
- 37.Hordijk, W. (1997) Correlation analysis of the synchronising-ca landscape. Physica D. 107, 255–264CrossRefGoogle Scholar
- 38.Hamming, R.W. (1980) Coding and Information Theory. Prentice-Hall, Inc., Englewood Cliffs, NJGoogle Scholar
- 39.Mohar, B. (1997) Some applications of laplace eigenvalues of graphs. In Hahn, G., Sabidussi, G., eds.: Graph Symmetry: Algebraic Methods and Applications. Volume 497 of NATO ASI Series C. Kluwer, DordrechtGoogle Scholar
- 40.Eiben, A.E., Bäck, T. (1997) Empirical investigation of multiparent recombination operators in evolution strategies. Evolutionary Computation. 5, 347–365CrossRefGoogle Scholar
- 41.Stadler, P.F., Seitz, R., Wagner, G.P. (1999) Evolvability of complex characters: Dependent fourier decomposition of fitness landscapes over recombination spaces. Bull. Math. Biol. Santa Fe Institute Report 99-01-001.Google Scholar
- 42.Spitzer, F. (1976) Principles of Random Walks. Springer-Verlag, New York, NYCrossRefGoogle Scholar
- 43.Priestley, M.B. (1981) Spectral Analysis and Time Series. Academic Press Inc., London, UKzbMATHGoogle Scholar
- 44.Weinberger, E.D. (1991) Fourier and taylor series on fitness landscapes. Biological Cybernetics. 65, 321–330zbMATHCrossRefGoogle Scholar
- 45.Reidys, C.M., Stadler, P.F.(1998) Neutrality in fitness landscapes. Technical Report 98-10-089, Santa Fe Institute Submitted to Appl. Math.& Comput. Google Scholar
- 46.Stadler, P.F. (1999) Spectral landscape theory. In Crutchfield, J.P., Schuster, P., eds.: Evolutionary Dynamics — Exploring the Interplay of Selection, Neutrality, Accident and Function. Oxford University Press, New York, NY (1999) To appear.Google Scholar
- 47.Kauffman, S., Levin, S. (1987) Towards a general theory of adaptive walks on rugged landscapes. J. Theor. Biol. 128, 11–45MathSciNetCrossRefGoogle Scholar
- 48.Box, G., Jenkins, G. (1970) Time Series Analysis, Forecasting and Control. Holden DayGoogle Scholar
- 49.Hordijk, W. (1996) A measure of landscapes. Evolutionary Computation. 4, 335–360CrossRefGoogle Scholar
- 50.Vassilev, V.K., Miller, J.F., Fogarty, T.C. (1999) Digital circuit evolution and fitness landscapes. In: Proceedings of the Congress on Evolutionary Computation Volume 2., IEEE Press, Piscataway, NY, 1299–1306Google Scholar
- 51.Vassilev, V.K., Miller, J.F., Fogarty, T.C. (1999) On the nature of two-bit multiplier landscapes. In Stoica, A., Keymeulen, D., Lohn, J., eds.: Proceedings of the 1st NASA/DoD Workshop on Evolvable Hardware, Los Alamitos, CA, IEEE Computer Society, 36–45CrossRefGoogle Scholar
- 52.Vassilev, V.K., Miller, J.F., Fogarty, T.C. (1999) Digital circuit evolution: The ruggedness and neutrality of two-bit multiplier landscapes. In Harvey, D.M., ed.: Evolutionary Hardware Systems, IEE Press, London, 6/1-6/4Google Scholar
- 53.Miller, J.F., Job, D., Vassilev, V.K. (2000) Principles in the evolutionary design of digital circuits — part II. J. Genetic Programming and Evolvable Machines 1 In press.Google Scholar
- 54.Miller, J.F., Thomson, P., Fogarty, T. (1997) Designing electronic circuits using evolutionary algorithms. arithmetic circuits: A case study. In Quagliarella, D., Periaux, J., Poloni, C., Winter, G., eds.: Genetic Algorithms and Evolution Strategies in Engineering and Computer Science. Wiley, Chechester, UK, 105–131Google Scholar
- 55.Andrews, P.B. (1990) An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof. Academic Press, Orlando, Florida (1986)Google Scholar
- 56.Chen, X., Hurst, S.L. (1982) A comparison of universal-logic-module realizations and their application in the synthesis of combinatorial and sequential logic networks. IEEE Transactions on Computers. C-31, 140–147CrossRefGoogle Scholar
- 57.Stadler, P.F., Grünter, W. (1993) Anisotropy in fitness landscapes. J Theor Bio. 165, 373–388CrossRefGoogle Scholar
- 58.Slavov, V., Nikolaev, N. (1999) Genetic algorithms, fitness sublandscapes and subpopulations. In Reeves, C., Banzhaf, W., eds.: Foundations of Genetic Algorithms, Volume 5. Morgan Kaufmann, San Francisco, CA, 199–218Google Scholar
- 59.Stadler, P.F., Happel, R. (1999) Random field models for fitness landscapes. J. Math. Biol. 38, 435–478MathSciNetzbMATHCrossRefGoogle Scholar
- 60.Burden, R.L., Faires, J.D. (1997) Numerical Analysis. Brooks/Cole Publishing Company, Pacific Grove, CA sixth edition.Google Scholar
- 61.Soderland, S., Fisher, D., Aseltine, J., Lehnert, W. (1995) Crystal: Inducing a conceptual dictionary. In: Proceedings of the 14th International Joint Conference on Artificial Intelligence, Morgan Kaufamnn, San Francisco, CAGoogle Scholar
- 62.Miller, J.F. (1999) An empirical study of the efficiency of learning boolean functions using a cartesian genetic programming approach. In Banzhaf, W., Daida, J., Eiben, A.E., Garzon, M.H., Honavar, V., Jakiela, M., Smith, R.E., eds.: Proceedings of the 1st Genetic and Evolutionary Computation Conference. Volume 2., Morgan Kaufmann, San Francisco, CA 1135–1142Google Scholar
- 63.Schwefel, H.P.(1981) Numerical Optimization of Computer Models. John Wiley & Sons, Chichester, UKGoogle Scholar
- 64.Bäck, T., Hoffmeister, F., Schwefel, H.P. (1991) A survey of evolutionary strategies. In Belew, R., Booker, L., eds.: Proceedings of the 4th International Conference on Genetic Algorithms, Morgan Kaufmann, San Francisco, CA 2–9Google Scholar
- 65.Ohta, T. (1992) The nearly neutral theory of molecular evolution. Annual Review of Ecology and Systematics. 23, 263–286CrossRefGoogle Scholar
- 66.Ohta, T. (1996) The current significance and standing of neutral and nearly neutral theories. BioEssays. 18, 673–684CrossRefGoogle Scholar
- 67.Huynen, M.A., Stadler, P.F., Fontana, W. (1996) Smoothness within ruggedness: The role of neutrality in adaptation. Proceedings of the National Academy of Science U.S.A. 93, 397–401CrossRefGoogle Scholar
- 68.Huynen, M.A. (1996) Exploring phenotype space through neutral evolution. Journal of Molecular Evolution. 43, 165–169CrossRefGoogle Scholar
- 69.Banzhaf, W. (1994) Genotype-phenotype-mapping and neutral variation — a case study in genetic programming. In Davidor, Y., Schwefel, H.P., Männer, R., eds.: Parallel Problem Solving from Nature III, Berlin, Springer-Verlag, 322–332CrossRefGoogle Scholar
- 70.Harvey, I., Thompson, A. (1996) Through the labyrinth evolution finds a way: A silicon ridge. In Higuchi, T., Iwata, M., Liu, W., eds.: Proceedings of the 1st International Conference on Evolvable Systems. Volume 1259 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 406–422Google Scholar
- 71.Gillespie, J.H. (1987) Molecular evolution and the neutral allele theory. In Harvey, P.H., Partridge, L., eds.: Oxford Surveys in Evolutionary Biology. Volume 4. Oxford University Press, New York, 11–25Google Scholar
- 72.Wolpert, D.H., Macready, W.G. (1997) No free lunch theorems for optimization. IEEE Transactions on Evolutionary Computation. 1, 67–82CrossRefGoogle Scholar
- 73.Biggs, N.J. (1995) Algebraic Graph Theory. Cambridge University Press, Cambridge, UK, second editionGoogle Scholar